Thanks for posting.
First, yes, you need a tapered leader. The tapering helps turn your fly over nicely -- the tapering should go from the approximate diameter of the end of your fly line to a fine tippet section to which you attach your fly. Manufactured tapered leaders are fine, but learn to replace the end section (tippet) by attaching suitable diameter leader material with a blood knot (many illustrations of this on the net). If you change flies often or break off on bushes and sixteen pound browns you didn't know lived in your small stream, you'll be wanting to replace the tippet section from time to time.
It's OK to cut off the loop if you want to. Attach the butt of the tapered leader to the fly line (here I'm assuming you have a double-tapered or weight forward tapered line that has a small diameter section just before the leader) with a nail knot (again, net). If you don't want to cut off the fly line loop, make a perfection loop knot at the end of your leader and join leader and line with a loop-to-loop connection.
A 3wt. system seems good for small steam fishing if your quarry is a foot long and down (that would be small in my humble opinion). If you bought a "package" (rod, reel and line) from a reputable dealer, it's most likely balanced and appropriate for what you describe.
Depending on how tight the holding spots are to reach and how close you must be to cover them, a nine-foot leader might work in some places, but a 7.5 foot leader might be better in others. If you loop-to-loop the leader to the line, it's much easier to change if you need to.
For lake fishing for small rainbows: if fish are not rising to the surface, you'll most likely want to be fishing nymphs or other types of wet-flies below the surface. You do not necessarily need a sinking line to do this effectively, but you might want to look into getting ahold of a slow-sinker if you do a lot of lake fishing.
Get some caddis, mayfly-nymph, and leech imitations in small sizes (No. 16 to maybe No. 10 or 8) some lightly weighed, others not.
How to fish these is totally dependent on the prevailing conditions. If someone is catching fish and is a nice gal or guy, don't hesitate to ask "what depth", "what fly", and "what retrieve." If no one is around, and you're sure there are catchable fish in the area, experiment among these three variables until your small rainbow latches on. Find out from a local fly shop what the favored patterns for any particular lake you're fishing. If you're using a recommended fly, then it cuts down to just two variables.
Hope this helps.
Totally changing the subject: have you fly-fished for winter steelhead? Just curious.